I may have to ban myself from growing trombas next year - could not keep up with how productive they were. Utterly dominated all my other squashes, now I have about 5 fully grown 4kg specimens I have no idea what to do with...
They store for a while as Mum says, but not as long as some squash. You can cut into chunks and roast with seasoning, herbs or spices. The roasted squash freezes well. You can make soup, add to casseroles or make a curry with it.
If you make soup, try grating in fresh root ginger to chopped onion and garlic. Cook down a little, add stock (from stock cubes is fine), seasoning and squash (pre-roasted or raw), simmer and blend. Roasted squash and homemade stock make it a luxury soup, but the basic version is good as well. Glorious colour, tastes great, does you the power of good